Well ... I took a punt, and ordered the 1840 birth certificate for Eliza Abbott registered in Shoreditch - and it is indeed her! She is illegitimate and her mother's maiden name is Wood.
Born: 19 Apr 1840 (baptism record says 29th) at 47 [Moneyew??] Street Hoxton New Town
Father: Jabez Abbott
Mother: Eliza Wood
Rank or Profession of Father: Farmer
Informant: Eliza Wood, mother, (24) [47 Moneyew Street (crossed out)] Woods Stables Colchester Street Whitechapel
When registered: 18 May 1840 (the day after the baptism)
In the right margin are words "Twenty" and a chopped off word above the signature of the registrar - a correction to the birth date perhaps?
I can no longer assume she died in infancy - in fact, I have a potential record in the 1841 census for her and her mother; but I now dont need to continue the search for a potential marriage for Jabez Abbott.
This is the first civil birth certificate I have seen for an illegitimate child. The index search result is misleading. Whether or not the parents are married, Wood is still the mother's maiden name.